A PATTERN FORMATION PROBLEM ON THE SPHERE. 1. Introduction

Transcription

1 Sixth Mississippi State Conference on Differential Equations and Computational Simulations, Electronic Journal of Differential Equations, Conference 5 007), pp ISSN: URL: or ftp ejde.math.txstate.edu login: ftp) A PATTERN FORMATION PROBLEM ON THE SPHERE C.E. GARZA-HUME, PABLO PADILLA Abstract. We consider a semi-linear elliptic equation on the sphere R n+ with n odd and subcritical nonlinearity. We show that given any positive integer k, if the exponent p of the nonlinear term is sufficiently close to the critical Sobolev exponent p, then there exists a positive solution with k peaks. Moreover, the minimum energy solutions with k peaks are such that the centers of these concentrations converge as p p to the solution of an underlying geometrical problem, namely, arranging k points on so they are as far away from each other as possible. We consider the equation. Introduction v + dn) + λ)v = v p,.) where is the Laplace-Beltrami operator on the sphere, n 3 and odd, < p < p = n + )/n ), p the critical Sobolev exponent, dn) = nn )/4 and λ > 0. We will write p = p ɛ. There are several motivations for studying this equation coming from geometry, analysis, mathematical biology, physics, etc. For p = p and λ = 0 equation.) is related to the Yamabe problem and has been extensively studied see for instance [4]). It is well known that in this case the associated variational problem exhibits lack of compactness and that the Palais-Smale PS) condition does not hold [9],[4]). Several authors have handled this difficulty by considering existence and blow up of solutions when p p [0]). In applications, equation.) appears also in several contexts related to pattern formation. It is obtained when looking for steady state solutions of the reduced Gierer-Meinhardt system when the diffusion rate of the activator to that of the inhibitor tends to zero, which is known as the shadow equation []. Similar models appear in other contexts, such as flame propagation see [5]). Considering the problem on the sphere is also natural from several perspectives. Usually, pattern formation models are studied on plane domains with boundary; however, there are several organisms, for instance radiolarians, that exhibit nearly) 000 Mathematics Subject Classification. 35B33, 35J0. Key words and phrases. Semilinear elliptic equation; sphere packing; critical Sobolev exponent; pattern formation. c 007 Texas State University - San Marcos. Published February 8,

2 98 C. E. GARZA-HUME, P. PADILLA EJDE/CONF/5 spherical symmetry. Some specimens of radiolarians show peaks similar to the ones depicted in figure. This was one of the main motivations for this work. In fact, it is perhaps more natural to visualize the dermis of an organism, on which the pattern formation process is taking place, as a surface, rather than a plane domain. It has been proposed that curvature effects can also be important in the selection of patterns and this makes it necessary to study the usual models on surfaces [3]). From the technical point of view, considering the sphere actually simplifies the problem, since boundary effects do not have to be taken into account. We can state our results in an informal way as follows for a precise formulation see section 3): there are solutions which exhibit a prescribed number of peaks. Moreover, among solutions with a fixed number of concentrations peaks), those with minimal energy are such that the centers of the concentrations tend to be as far as possible from each other, thus solving a sphere packing problem. A very similar result has been obtained by Gui and Wei [8] for domains in R n with boundary. There are however several differences. First of all, these authors consider the equation ɛ u u + fu) = 0, u > 0 in Ω, u ν = 0 on Ω and their analysis is made taking ɛ as a small parameter. Moreover, as stated above, the presence of the boundary makes some of the estimates and technicalities somewhat involved. On the other hand, we take the coefficient ɛ of the Laplacian as fixed, and vary the exponent of the nonlinear term. Although these approaches are in some sense equivalent, as has been shown by Benci and Cerami see for instance [6]), considering the exponent as our parameter allows us to use well known global compactness results see [3] or [4]). The existence of k-peaked solutions as defined in section 3) is obtained by minimization on a suitable space of functions, basically, those with the right π/ksymmetry. We will construct an action that does not have fixed points on the sphere. Therefore, if a solution has a peak, by symmetry it has to have at least k. From the mathematical point of view, it is known that if λ = 0 and p = p there is a unique nontrivial positive solution up to rotations and scalings), in other words, for the critical case, there are no k-peaked solutions on the sphere. Roughly speaking, this is due to the fact that the Palais-Smale condition is not valid any more and PS) sequences that exhibit more than one peak are not compact. Using Bahri s terminology, these constitute critical points at infinity [3]). In our case, the fact that the exponent of the non linear term remains strictly smaller than the critical exponent implies that the PS) condition still holds and therefore PS) sequences with k peaks do have subsequences converging to actual solutions. The analysis of existence and blow up when the linear term is not present has been the subject of much research; see for instance the work by Y. Li [0], and references therein. In order to make the reasoning more transparent, we include in section 3 an example in which the existence of multipeak solutions is more easily established. Namely, we look for solutions with two peaks. In this case, constraining the functional to even functions on the sphere is sufficient, since the action of the group, in this case the antipodal action, has only one fixed point, the origin, which does not belong to the sphere.

3 EJDE-006/CONF/5/ A PATTERN FORMATION PROBLEM 99 The rest of the paper is organized as follows. In section we formulate the variational problem and recall some well known facts. In section 3 we first state our results: existence of k-peaked solutions theorem 3. and remark 3.), and their geometric properties theorem 3.). In section 3. we give the proofs. We conclude in section 4 with some questions and open problems.. Formulation of the problem We can work directly on or use stereographic projection and work on R n, but generally computations are easier on R n and many standard results are stated in that setting. Let P = 0,..., 0, ) be the north pole on R n+. Stereographic projection σ : {P } R n is defined by σx,..., x n, ξ) = y,..., y n ) for x, ξ) {P } where y j = x j / ξ). We recall that equation.) with p = p ɛ is transformed, after stereographic projection on R n, into 4λ n ) ) ɛ u + + y u = n ) ) 4n ) + y u p..) Note that the term containing dn) cancels out. This equation is equivalent to.) in the sense that if v is a solution of.) and ) n vx) = + y uy) with y = yx)) then u solves equation.) see [9]). Associated with these two equations we have the following functionals ES ɛ nv) = dn) + λ) v + v + ɛ ) vp p dσ + ɛ in H ) and E ɛ u) = u + R n λ + y ) u n ) ) ɛ n ) ɛ+ ) up 4n ) + y p dy ɛ + in H R n ). Note that for small ɛ > 0 it is a standard fact that the Palais-Smale PS) condition holds for ES ɛ and for n Eɛ when restricted to bounded domains [4]) and solving equations.) or.) is equivalent to finding critical points of the corresponding functionals in the following sense. The functional ES ɛ n is not bounded below but one can add a constraint to be able to apply the direct method of the calculus of variations as done in [[4], I.]. The idea is to consider the functional v + dn) + λ) v ) dσ.) restricted to the set where v p + ɛ =. Using Lagrange multipliers this yields the functional S defined by: [ dn) + λ) S = S v + v ]/ ) /p + ɛ) v p + ɛ..3) n

4 00 C. E. GARZA-HUME, P. PADILLA EJDE/CONF/5 We will use a modification of a global compactness result by Struwe which we quote for completeness. For a bounded domain Ω R n define E λ = u + λ u ) dx u dx, Ω = n/n ) = p + and let D, Ω; R n ) be the completion of C 0 Ω; R n ) in the norm u D, = u L. Theorem. [4, Thm. III. 3. p. 69]). Suppose Ω is a bounded domain in R n, n 3 and for λ R let u m ) be a Palais-Smale sequence for E λ in H, 0 Ω) D, R n ). Then there exists an index k N 0, sequences Rm), j x j m), j k, of radii Rm) j m ) and points x j m Ω, a solution u 0 H, 0 Ω) D, R n ) to the problem u = λu + u u in Ω u > 0 in Ω, u = 0 on Ω and non-trivial solutions u j D, R n ), j k, to the limiting problem u = u u such that a subsequence u m ) satisfies um u 0 Here u j m denotes the rescaled function Moreover k j= in R n, Ω.4) u j m D, 0..5) R n ) u j mx) = R j m) n u j R j mx x j m)), j k, m N. E λ u m ) E λ u 0 ) + k E 0 u j ). Remark.. An analogous result when Ω is a compact subset of different from ) can be found in [3] or [4]. We can in fact apply the result to in this particular case because a minimizing sequence cannot blow-up at an infinite number of points on the sphere. Clearly any sequence with a finite number of blowup points would have less energy. Therefore we can rotate so that the north pole is not a blow-up point. We can choose a ball B around the north pole where we can apply lemma 3.4 in []: Lemma.3 [, lemma 3.4]). Let Ω m = {x R n λ / m x Ω}, Ω a bounded open set containing the origin, with λ m as m. Suppose v m ) m N W, Ω m ) is a sequence with v m W, Ω m) uniformly bounded. If for any R > 0, ) then lim m sup y R n Ω m B R y) j= v m p+ dx lim m Ω m v m p+ dx = 0. = 0,

5 EJDE-006/CONF/5/ A PATTERN FORMATION PROBLEM 0 The proof of the above lemma can be found in [5]. Since there is no blow-up in B, the condition of the lemma is satisfied and so u m 0 strongly in L p+ Ω). By considering the complement B ) c, where B is a slightly smaller ball around the north pole, we can substitute the condition u m H, 0 Ω) in Theorem. by the fact that in a neighborhood of the boundary u m 0 in the L p+ norm. 3. Main results We begin by stating our results in a precise way. Theorem 3.. Let λ > 0 and k N + be given. Then there exists an ɛ 0 > 0 depending on λ and k such that equation.) on, n 3 and odd with p = p ɛ has a positive solution, u ɛ, for all 0 < ɛ ɛ 0, which concentrates at k different points x j, j =,... k. Remark 3.. That a positive solution, u ɛ, 0 < ɛ ɛ 0 concentrates at k different points x j, j =,... k, or that u ɛ has k peaks means that when ɛ 0, after rotations and rescalings, a subsequence converges strongly in H ) to a linear combination of k distinct solutions of the limiting problem Cf. Theorem., global compactness result.) v + dn)v = v p. 3.) Theorem 3. states that there is at least one solution with k peaks. Now, for any fixed, small, positive ɛ the Palais-Smale condition is satisfied. Then, since solutions with exactly k peaks and bounded energy form a compact set, we can assume that there is a solution u ɛ, not necessarily the same one found in theorem 3.) that, among all the k-peaked solutions, has least energy S. We now show that the behavior of the k concentrations for u ɛ when ɛ 0 is governed by an underlying geometric problem: Theorem 3.3. Let v ɛi, with ɛ i 0 as i be a sequence of k peaked solutions of equation.) which have least energy S as discussed in the preceding paragraph and let x j i, j =,... k, i =,,... be the center of the jth peak. Then there exists a subsequence still denoted x j i ) such that xj i xj, where x j is a solution of the following maximization problem: max x j x l x j,x l where x j x l if j l. j l Remark 3.4. From theorem 3. it follows that the concentrations of least energy k peaked solutions, as ɛ i 0, arrange themselves according to a packing problem on the sphere in the sense that the centers of the concentrations peaks) tend to be as far away from each other as possible. Proofs We begin by proving Theorem 3., the existence of k peaked solutions. This is done by minimizing on a suitable set the functional [ dn) + λ) S v) = S v + v ]/ ) /p + ɛ) v p + ɛ. n

6 0 C. E. GARZA-HUME, P. PADILLA EJDE/CONF/5 We will restrict to functions which are invariant under the action of g given by 3.) below. Case k =. Before proving theorem 3. in the general case, we mention that the case k =, namely existence of a solution with two peaks can easily be established. In fact, it is sufficient to minimize the corresponding functional, S, in the subspace of even functions, that is, functions invariant under the antipodal action x x. Then by direct minimization and the fact that the PS) condition holds for < p = p ɛ < p, we obtain a solution of.). Now, by continuity of the L p norm with respect to p, a sequence of such solutions, u ɛm, with ɛ m 0 as m, also forms a PS) sequence for problem.). We can now apply Theorem. as explained in remark.. In particular, there has to be at least one solution, u, of the limiting problem mentioned in the theorem with the corresponding sequences R m and centers x m as in.3). But due to the symmetry of the functions we are considering, we see that in fact there has to be at least two sequences of points of concentration, namely, x m and x m. So for ɛ sufficiently small i sufficiently big), the solution looks like the one depicted in figure. Figure. Sketch of solution with two peaks Proof of 3.. We now give the proof for the general case. In fact, the reasoning given above for the antipodal action can be applied, provided we can construct an action with no fixed points on the sphere and such that any orbit has k distinct points. Since n is odd we can write the action g in R n+ in polar coordinates as R θ 0 R θ ) cos θ sin θ g =..., R θ = 3.) sin θ cos θ 0 R θ where θ = π/k. This action has no fixed points on the sphere. As before, we minimize S in the subspace X H of functions invariant under g. The fact that a critical point exists is standard. The existence of k concentrations for ɛ sufficiently small follows as in the case k =. When n is even we cannot construct an action without fixed points on the sphere that leaves the Laplacian invariant.

7 EJDE-006/CONF/5/ A PATTERN FORMATION PROBLEM 03 Figure sketches the solution with six peaks. Figure. Sketch of solution with six peaks Proof of 3.3. It remains to be shown that the centers of the concentrations satisfy a sphere-packing problem, that is, they are as far away from each other as possible. This is based on the expansion of the energy in terms of the energies of the separate peaks. For a similar expansion see [3] or [8]. Let v ɛi, ɛ i 0 be a sequence of minimizers as in the statement of the theorem. Single-peaked solutions to the limiting problem, after stereographic projection, have the form [4],Ch.III sect. ): w j = t j ) ) n )/ = t j ) + y y j /t j ) + y yj t j ) ) ) n )/. 3.3) These solutions are radially symmetric with respect to y j, where the maximum is attained and they concentrate more as t 0. We will call w j x) the corresponding single-peaked solutions on the sphere see [3]). In S 3 they are given by /t w j j x) =, t j ) + + t j ) ) cos dx, x j ) where dx, x j ) is the geodesic distance on the sphere. By the global compactness characterization of PS sequences the behavior of the energy.3) is given to leading order by the expression: I ɛ = V m ) + d + λ)vm dσ V p + ɛ, 3.4) m dσ) p + ɛ where V m = k j= wj m. We can assume without loss of generality that t j m = t m = min j {t j m} since this decreases I ɛ in 3.4). The following estimate holds:

8 04 C. E. GARZA-HUME, P. PADILLA EJDE/CONF/5 Proposition 3.5. Let δ > 0 be given. Then there exists t 0 δ) and C n, k), C n, k), e, e all greater than zero such that k I ɛ = C C y j y l + e + e j l where e 0 when ɛ 0, e < δ for all t 0 > t. Proof. First observe that, in the limiting case, the quantity J = w j dσ )/ w j p does not depend on either t or y j. Indeed, J = /t R n R n /t z z j + t + dσ) /p +) ) n )/ Jac dz + z zj t ) ) ) n )/ n/n ) ) n )/n 3.5) Jac dz and making the change of variables y = z zj t expression 3.5) becomes R + y )n )/ / ) ) nt n )/n t tn Jac dy n n R + y Jac dy n / = R + y )n )/ ) ) n n )/n Jac dy Jac dy, n R + y n which is independent of t and y j. By a similar change of variables we have that λ / /p wj ) dσ w j ) dσ) p + ɛ + ɛ) tends to zero with t uniformly in 0 ɛ because of the factor t n n/p + ɛ) which appears after scaling and has a positive exponent if ɛ < 4/n ). We can take t small enough so that the term containing V m in 3.4) is small, say less than ɛ. Recall that the L p norm is an increasing and continuous function of p and so, for ɛ small we can write for any v L p +, v L p + ɛ = v L p + p ɛ), where p ɛ) 0 and tends to zero with ɛ. It also follows from Minkowski s inequality that /p dσ) + +) ) = w j p /p +) + p t), v p j where vx) = vσ y)) = V y) and p t) 0 and tends to 0 with t. So we can choose t 0 sufficiently small such that for all ɛ < ɛ 0 and t < t 0 3.4) becomes IS ɛ = v + ɛ dσ n k w S j ) n p + /p +) dσ P = k w j p + dσ ) /p +) v + ɛ dσ P/ k w S j n p + dσ ) /p +) [ where ɛ, P 0 and small. Expanding v and using the fact that to first order = + a + higher order terms a ],

9 EJDE-006/CONF/5/ A PATTERN FORMATION PROBLEM 05 we obtain the expansion I ɛ = C + C j,l w j w l + Q, where C, C depend only on k and n and Q is small for t < t 0 uniformly in ɛ < ɛ 0. To calculate the integral w j w l 3.6) we can, without loss of generality, rotate and assume that the centers x j, x l lie on the unit circle on the horizontal plane. We can now do stereographic projection and use expression 3.3). However, w j w l = n R n ) R n /t + y yj t ) n 4 /t )y yj ) t + y yj t ) )y yl ) t + y yl t ). The leading order in t this is proportional to + y yl t ) n 4 R n y y j ) y y l ) dx. 3.7) In this expression, the term containing y gives a fixed contribution. Those corresponding to y j y vanish since they are odd functions integrated over a domain symmetric with respect to the origin. The term containing y j y l can be written as C y j y l cos α jl, where α jl is the angle between y j and y l. By construction y j = y l = and α jl is proportional to the geodesic distance on the sphere, dx j, x l ). So minimizing w j w l 3.8) j,l is equivalent to minimizing j,l cosα j,l). Thus the minimum energy for k peaked solutions is achieved when the distance among the k peaks is maximized. This proves the proposition. For similar computations see[3]). 4. Open problems Some of the results that we use were originally proved for n 3 but similar ones have been shown to hold for n = []) so in principle we could also handle the case n = which is more directly related to biological problems if the parity issue is resolved. It would be interesting to study pattern formation on general surfaces, not just the sphere and to analyze the way in which the geometry affects the location of peaks. Another interesting question is the stability of the multipeak solutions and the dynamics of the problem under the negative gradient flow. On the other hand, similar pattern formation problems on surfaces, but with different nonlinearities, for instance, bistable potentials, also appear very often and are the object of current research.

10 06 C. E. GARZA-HUME, P. PADILLA EJDE/CONF/5 Acknowledgments. We would like to thank Dr. A. A. Minzoni for many useful discussions and suggestions and Dr. A. Castro for many improvements in the manuscript. This work was supported in part by Conacyt Group Project U F and project 3403-E. We thank Ana Pérez Arteaga for computational support. Part of this work was carried out while the authors were visiting the Mathematical Institute, Oxford University, UK. References [] Adimurthi, Struwe, M., Global compactness properties of semilinear elliptic equations with critical exponential growth, J. Funct. Analysis ), [] Aubin, T., Nonlinear Analysis on manifolds. Monge-Ampère Equations, Springer-Verlag, New York, Heidelberg, Berlin 98. [3] Bahri, A., Critical points at infinity in some variational problems, Pitman Research Notes in Mathematics 8, Longman Scientific and Technical, UK 989. [4] Bahri, A., Coron, J.M., On a Nonlinear Elliptic Equation Involving the Critical Sobolev Exponent: The Effect of the Topology of the Domain CPAM 4 988, pp [5] Berestycki, H., Nirenberg, L., Traveling fronts in cylinders, Annales de l IHP, Analyse non linéare, Vol.9, 99, no.5, pp [6] Benci, V.; Cerami, G., Positive solutions of some nonlinear elliptic problems in exterior domains, Arch. Rational Mech. Anal ), no. 4, [7] Gilbarg,D., Trudinger, N.S., Elliptic Partial Differential Equations of Second Order, Springer-Verlag, 983. [8] Gui, C., Wei, J., Multiple interior peak solutions for some singularly perturbed Neumann problems, J. Differential Equations, ), no., 7. [9] Lee, J.M.; Parker, T.H. The Yamabe problem. Bull. Amer. Math. Soc. N.S.) 7 987), no., [0] Li, Y., Prescribing scalar curvature on and related problems II, Communications in Pure and Applied Math, ) no. 6. [] Maier-Paape, S.; Schmitt, K.; Wang, Z-Q., On Neumann Problems for semilinear elliptic equations with critical nonlinearity: existence and symmetry of multipeaked solutions, Comm. in PDE,, pp , 997. [] Ni, Wei-Ming; Takagi, I., On the Neumann problem for some semilinear elliptic equations and systems of activator-inhibitor type, Trans. Amer. Math. Soc ), no., [3] Padilla,P., Plaza, R., Sanchez, F., Barrio, R., Maini, P. The effect of growth and curvature on pattern formation, J. Dyn. Diff. Eqns. 6 No. 4, Oct [4] Struwe, Michael, Variational Methods: Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems, Springer-Verlag Berlin, Heidelberg, 990. [5] Wang, Z-Q., The effect of the domain geometry on the number of positive solutions of Neumann problems with critical exponents, Diff. Integral Equations, 8, pp , 995. Clara E. Garza-Hume IIMAS-FENOMEC, Universidad Nacional Autónoma de México, Circuito Escolar, Cd. Universitaria 0450, México D. F., México address: clara@mym.iimas.unam.mx Pablo Padilla IIMAS-FENOMEC, Universidad Nacional Autónoma de México, Circuito Escolar, Cd. Universitaria 0450, México D. F., México address: pablo@mym.iimas.unam.mx